Systems formed by translates of one element in $L_p(\mathbb R)$

Abstract

Let $1\le p <\infty$, $f\in L_p(\real)$ and $\Lambda\subseteq \real$. We consider the closed subspace of $L_p(\real)$, $X_p (f,\Lambda)$, generated by the set of translations $f_{(\lambda)}$ of $f$ by $\lambda \in\Lambda$. If $p=1$ and $\{f_{(\lambda)} :\lambda\in\Lambda\}$ is a bounded minimal system in $L_1(\real)$, we prove that $X_1 (f,\Lambda)$ embeds almost isometrically into $\ell_1$. If $\{f_{(\lambda)} :\lambda\in\Lambda\}$ is an unconditional basic sequence in $L_p(\real)$, then $\{f_{(\lambda)} : \lambda\in\Lambda\}$ is equivalent to the unit vector basis of $\ell_p$ for $1\le p\le 2$ and $X_p (f,\Lambda)$ embeds into $\ell_p$ if $2<p\le 4$. If $p>4$, there exists $f\in L_p(\real)$ and $\Lambda \subseteq \zed$ so that $\{f_{(\lambda)} :\lambda\in\Lambda\}$ is unconditional basic and $L_p(\real)$ embeds isomorphically into $X_p (f,\Lambda)$.